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I am studying the nonlinear PDE: $$ \frac{\partial}{\partial t} f(\boldsymbol{x},t) = \left\| \frac{\partial}{\partial \boldsymbol{x}} f(\boldsymbol{x},t) \right\| $$ where $\boldsymbol{x} \in \mathbb{R}^n$, $t \in \mathbb{R}^+$, and $f:\mathbb{R}^n \times \mathbb{R}^+ \rightarrow \mathbb{R}$. Here $\| \cdot \|$ denotes the standard $L_2$ norm of a vector. This PDE is of course accompanied by a given initial condition $f(\boldsymbol{x},t=0)=f_0(\boldsymbol{x})$ and some boundary conditions (either Dirichlet or Neumann is fine).

The equation is nonlinear, so I have little hope for a closed form solution, but I was hoping that I could at least find some references that study the properties of this PDE and its solution. I went through multiple resources that enlist nonlinear PDEs, and surprisingly I was not able to find any reference to such a simple equation.

Any pointer to the name of this PDE (so that I can look it up myself and learn more about this PDE), or any material that explains the properties of the solution of this PDE, would be greatly appreciated.

Thanks!

Golabi

user407223
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  • I believe I can offer a solution. Would you like to see it? (If I'm not overlooking something, the solution is not incredibly interesting, and maybe that could explain it if no name has been given to this equation.) – 2'5 9'2 Mar 29 '22 at 04:48
  • Definitely interested! I can see that the limit case $t \rightarrow \inf$ is not that interesting, but I would really like to know the intermediate solution forms i.e. when $0<t<\infty$. Thanks! – user407223 Mar 29 '22 at 05:01
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    Choose $c_i$ and $k$. Consider $f(x,t)=\exp\left(\sum c_ix_i+t\sqrt{\sum c_i^2}+k\right)$. – 2'5 9'2 Mar 29 '22 at 05:03
  • Thank you Alex, but this solution only works assuming the initial condition is f(x,0)=exp(k+\sum_i c_i x_i). I am interested a general solution that holds for any provided initial condition function f(x,0). Maybe this was not clear in the post... I made some edits to hopefully make it clearer. – user407223 Mar 29 '22 at 05:12
  • I see. Yes, I was missing something. – 2'5 9'2 Mar 29 '22 at 05:26
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    This is a Hamilton-Jacobi equation $u_t +H(\nabla u ,x) =0$ with Hamiltonian $H(p,x) =- \vert p \vert $, $(p,x)\in \mathbb R^n \times \mathbb R^n$ . – JackT Mar 29 '22 at 05:47
  • Thanks! This was very helpful. Is the choice of the Hamiltonian function $H(p,x)=-|p|$ something that has been of interest and/or studied before? – user407223 Mar 29 '22 at 05:53

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